Someone I know spells it "Gruess Gott" and explained the "ue" is a way of indicating it is an umlauted "u" - Has anyone ever heard of this? Is it proper? (This is in a context where no special characters from German are available at the keyboard, US English only)
It's an "umlaut transliteration", if you will, and was quite common in the days of typewriters. In these days of ubiquitous Unicode, there's little excuse for it (US-keyboard or no). Here's probably more than you wanted to know: Conversion table for diacritics (e.g. "ü" → "ue")
If there is no way to type an umlaut character, then replacing e.g. ü by ue is the only correct option. Replacing it by u as you did in your question before the edit is incorrect. Indeed Gruß and grüß (or transliterated Gruss and gruess) are distinct German words with differently pronounced vowels. The latter is the imperative of the verb grüßen (to greet), the former is a noun derived from it. Replacing the latter by the former makes a sentence ungrammatical.
I should add that the etymology of Grüß Gott is a bit more involved, it is not a request to greet God as might seem from what I wrote above.
I should also note that the above is from a German language perspective. When rendering German words in other languages, as in English, different customs may have evolved.
The previous answers are all correct, I just wanted to add that the version "OE, AE, UE" instead of "Ö, Ä, Ü" is actually the older one. It used to be OE, which became contracted to Oͤ (an O with a small e), which later got simplified to Ö